Find the orthogonal trajectories of the following familiy of curves
$y = C e^{- 2 x}$

y^{2} = x + K

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y^{2} = - x + K

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y^{2} = x^{2} + K

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y^{2} + x^{2} = K

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Solution

The correct option is A $y^{2} = x + K$
$y = c e^{- 2 x} \Rightarrow y e^{2 x} = c \Rightarrow y^{'} e^{2 x} + 2 y e^{2 x} = 0 \Rightarrow y^{'} = - 2 y$
Replacing $\frac{d y}{d x}$ by $\frac{- d x}{d y}$ , we get
$y^{'} = \frac{1}{2 y} \Rightarrow 2 y d y = x d x \Rightarrow y^{2} = x + K$

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